Data collection, reduction, and analysis

Other popular methods for numerical solutions of DEs are the Runge-Kutta methods. They again come in forms of different order, depending on the number of selected points on each sub-interval for which the function is evaluated and averaged. The development of these methods includes quite sophisticated analyses of errors (deviations from the true solutions) which occur with functions of different properties. A major problem in the numerical integration of rate equations is stiffness. A differential equation is called stiff if, for instance, different st s in the process occur on widely different time scales. It is very in dent to compute with time intervals suitable for the steepest part of the progress curve (see Press et al., 1986, chapter 16 and commercial programs recommended on p. 36). 

The use of numerical integration for simulating reaction mechanisms makes a great contribution to the study of complex systems. It should, however, be applied with a scalpel rather than a sledge hammer. Whenever possible analytical equations should be obtained, at least for limiting cases, and compared with numerical solutions. When this is not possible it is advisable to write out the differential equations rather than to rely entirely on a symbolic processor to convert a description of the mechanism. Saving time is not the most important aspect of any procedure involved in research. Some laborious occupations help one to think about the problem. 

A numerically specified function, that is a dataset, can almost always be made to fit one or more of the following sum of exponentials, power series or Fourier series regardless of its physical significance. Acton s (1970) chapter entitled What not to compute is essential reading. Ashe points out most of us can more easily compute than think . None the less, if correctly used, digital collection and computer analysis of data have brought many insoluble problems into the realm of the soluble. 

If one is investigating a reaction with a very complex mechanism, that is with many parameters in the rate equation, fitting may not give a unique solution. In such a case a useful exercise is to overlay a simulated curve over the experimental record, using estimated values for the parameters in the rate equation. Successive changes in the values for the parameters will lead to a potential solution. This sounds more tedious than it is and, furthermore, it is a useful exercise because it will also show which parameters have the greatest influence on the behaviour of the function and it makes one think about the problem at the same time. 

When the mechanism is very complex or non-linear (as defined in section 2.4 p. 42) it is not possible to obtain a satisfactory analytical solution and (as discussed above) a set of differential equations, one for each state of the system, is the only available description of the mechanism. In such cases one can proceed either by trial and error (the above mentioned overlay method), using numerical simulation of the differential equations instead of the analytical equation. Alternatively Scientist, unique among the programs mentioned above, has facilities for non-linear least square fitting of data which can only be described by sets of differential equations. Kinetic instrument manufacturers are, increasingly, making such programs commercially available. 

---

The nonconformity data should be collected and quantified using one of the seven quality tools (see Part 2 Chapter 14), preferably the Pareto analysis. You can then devise a plan to reduce the 20% of causes that account for 80% of the nonconformities. However, take care not to degrade other processes by your actions (see Theorg ofcon-staints in Part 2 Chapter 2). The plan should detail the action to be taken to eliminate the cause and the date by which a specified reduction is to be achieved. You should also monitor the reduction. The appropriate data collection measures therefore need to be in place to gather the data at a rate commensurate with the production schedule. Monthly analysis may be too infrequent analysis by shift may be more appropriate. 

Crystal data and details of data collection, data reduction and final refinement are reported in Table 1. The procedure for data collection and processing, which included a correction for scan-truncation effects, were similar to those recently described for syn-l,6 8,13-biscarbonyl[14]annulene [10] and citrinin [11], Figure 1 shows the numbering scheme adopted in the present analysis. 

It is interesting to trace the development of instrument automation over the relatively brief period of the past ten to fifteen years. Early in this period, a truly automated instrument was a rare and expensive item built around a costly dedicated minicomputer. Automated data collection and analysis from any instrument which was not automated at the factory was usually accomplished by digitizing the data and storing it on a transportable media such as paper tape. These data were then delivered and fed to a timeshare system of some sort on which the data reduction program ran and which printed a report and sometimes a plot of the data. Often a considerable time delay occured between the generation and the analysis of the data. The scientist was at the mercy of the computer elite who could implement his data logger and provide the necessary computer resources to analyze his data. The process was expensive, both in time and in money. 

Data are the raw product of the scientific method of inquiry. By analysis, refinement and reduction which collectively constitute the fourth step in the sequence, data are converted to information about the nature of study systems. The conversion is accomplished by the Neymann-Pearson process of statistical hypotheses testing(g,). If the collected data are sufficient and pertinent enough to support rejecting or accepting the statistical hypothesis under test, a measurableO. 10) quantity of information about the study system has been extracted. If not, the data cannot be converted to information and therefore cannot contribute to the pool of accepted scientific knowledge. 

When necessary, the assessor should consider the use of more advanced analysis methods (e.g. probability-based methods or more sophisticated/rigorous models) and focused data collection efforts to facilitate the development of a modeling approach that will result in estimates that are more representative of the actual exposure distribution. In the case of data collection, the value of information that could be gained is an important part of justification for the reduction of uncertainties by further study. 

The most important parameter is a clear identification of the specific question that the toxicity test is supposed to answer. The determination of the LC50 within a tight confidence interval will often require many fewer organisms than the determination of an effect at the low end of the dose-response curve. In multispecies toxicity tests and field studies, the inherent variability or noise of these systems requires massive data collection and reduction efforts. It is also important to determine ahead of time whether a hypothesis testing or regression approach to data analysis should be attempted. 

---